*Lower Division Courses*

**1A. Calculus. (4) **
Students will receive no credit for 1A after taking 16B and 2 units after taking 16A. Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic test, or 32. Consult the mathematics department for details. Students with AP credit should consider choosing a course more advanced than 1A.*
This sequence is intended for majors in engineering and the physical sciences. An introduction to differential and integral calculus of functions of one variable, with applications and an introduction to transcendental functions.
(F,SP)

**1B. Calculus. (4) **
Students will receive 2 units of credit for 1B after taking 16B. Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: 1A.*
Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.
(F,SP)

**H1B. Honors Calculus. (4) **
Students will receive 2 units of credit for H1B after taking 16B. Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: 1A.*
Honors version of 1B. Continuation of 1A. Techniques of integration; applications of integration. Infinite sequences and series. First-order ordinary differential equations. Second-order ordinary differential equations; oscillation and damping; series solutions of ordinary differential equations.
(F)

**10A. Methods of Mathematics: Calculus, Statistics, and Combinatorics. (4) **
Students will receive 2 units for 10A after taking 1A. Three hours of lecture and three hours of discussion per week.
*Prerequisites: Three and one-half years of high school math, including trigonometry and analytic geometry.*
This sequence is intended for majors in the life sciences. Introduction to differential and integral calculus of functions of one variable. Representation of data, elementary probability theory, statistical models, and testing.
(F,SP)
*Staff*

**10B. Methods of Mathematics: Calculus, Statistics, and Combinatorics. (4) **
Students will receive 2 units for 10B after taking 55. Three hours of lecture and three hours of discussion per week.
*Prerequisites: Continuation of 10A.*
Elementary combinatorics and discrete probability theory. Introduction to graphs, matrix algebra, linear equations, difference equations, and differential equations.
(F,SP)
*Staff*

**16A. Analytic Geometry and Calculus. (3) **
Students will receive no credit for 16A after taking 1A. Two units of 16A may be used to remove a deficient grade in 1A. Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional one hour to one and one-half hours of lecture or discussion/workshop per week.
*Prerequisites: Three years of high school math, including trigonometry, plus a satisfactory grade in one of the following: CEEB MAT test, an AP test, the UC/CSU math diagnostic exam, or 32. Consult the mathematics department for details.*
This sequence is intended for majors in the life and social sciences. Calculus of one variable; derivatives, definite integrals and applications, maxima and minima, and applications of the exponential and logarithmic functions.
(F,SP)

**16B. Analytic Geometry and Calculus. (3) **
Students will receive no credit for 16B after 1B, 2 units after 1A. Two units of 16B may be used to remove a deficient grade in 1A. Two hours of lecture and one hour of discussion/workshop per week; at the discretion of the instructor, an additional hour of lecture or discussion/workshop per week.
*Prerequisites: 16A.*
Continuation of 16A. Application of integration of economics and life sciences. Differential equations. Functions of many variables. Partial derivatives, constrained and unconstrained optimization.
(F,SP)

**24. Freshman Seminars. (1) **
Course may be repeated for credit as topic varies. One hour of seminar per week.
Sections 1-2 to be graded on a letter-grade basis. Sections 3-4 to be graded on a *passed/not passed* basis.
The Berkeley Seminar Program has been designed to provide new students with the opportunity to explore an intellectual topic with a faculty member in a small-seminar setting. Berkeley Seminars are offered in all campus departments, and topics vary from department to department and semester to semester.
(F,SP)

**32. Precalculus. (4) **
Students will receive no credit for 32 after taking 1A-1B or 16A-16B and will receive 3 units after taking 96. Two hours of lecture and two hours of discussion per week, plus, at the instructor's option, an extra hour of lecture/discussion per week.
*Prerequisites: Three years of high school mathematics, plus satisfactory score on one of the following: CEEB MAT test, math SAT, or UC/CSU diagnostic examination.*
Polynomial and rational functions, exponential and logarithmic functions, trigonometry and trigonometric functions. Complex numbers, fundamental theorem of algebra, mathematical induction, binomial theorem, series, and sequences.
(F,SP)

39A. . (2-4)
Two to four hours of seminar per week.

**49. Supplementary Work in Lower Division Mathematics. (1-3) **
Course may be repeated for credit. Meetings to be arranged.
*Prerequisites: Some units in a lower division Mathematics class.*
Students with partial credit in lower division mathematics courses may, with consent of instructor, complete the credit under this heading.
(F,SP)

**53. Multivariable Calculus. (4) **
Students will receive no credit for Mathematics 53 after completing Mathematics W53, 53M; 3 units for Mathematics 50A and 1 unit for Mathematics 50B. A deficient grade in 53 may be removed by completing Mathematics W53. Three hours of lecture and two hours of discussion per week.
*Prerequisites: Mathematics 1B.*
Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.
(F,SP)

**H53. Honors Multivariable Calculus. (4) **
Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: 1B.*
Honors version of 53. Parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. Theorems of Green, Gauss, and Stokes.
(F,SP)

**54. Linear Algebra and Differential Equations. (4) **
Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: 1B.*
Basic linear algebra; matrix arithmetic and determinants. Vector spaces; inner product as spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.
(F,SP)

**H54. Honors Linear Algebra and Differential Equations. (4) **
Three hours of lecture and two hours of discussion per week.
*Prerequisites: 1B.*
Honors version of 54. Basic linear algebra: matrix arithmetic and determinants. Vectors spaces; inner product spaces. Eigenvalues and eigenvectors; linear transformations. Homogeneous ordinary differential equations; first-order differential equations with constant coefficients. Fourier series and partial differential equations.
(F,SP)

**55. Discrete Mathematics. (4) **
Students will receive no credit for 55 after taking Computer Science 70. Three hours of lecture and two hours of discussion/workshop per week; at the discretion of the instructor, an additional hour of discussion/workshop or computer laboratory per week.
*Prerequisites: Mathematical maturity appropriate to a sophomore math class. 1A-1B recommended.*
Logic, mathematical induction sets, relations, and functions. Introduction to graphs, elementary number theory, combinatorics, algebraic structures, and discrete probability theory.
(F,SP)

**74. Transition to Upper Division Mathematics. (3) **
Three hours of lecture and two hours of discussion per week.
*Prerequisites: 53 and 54.*
The course will focus on reading and understanding mathematical proofs. It will emphasize precise thinking and the presentation of mathematical results, both orally and in written form. The course is intended for students who are considering majoring in mathematics but wish additional training.
(F,SP)

**91. Special Topics in Mathematics. (4) **
Course may be repeated for credit. Three hours of lecture/discussion per week.
Topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See department bulletins.
(F,SP)
*Staff*

**98. Supervised Group Study. (1-4) **
Must be taken on a *passed/not passed* basis.
Directed Group Study, topics vary with instructor.
(F,SP)

**98BC. Berkeley Connect. (1) **
Course may be repeated for credit. One hour of discussion per week.
Must be taken on a passed/not passed basis.
Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.
(F,SP)

**99. Supervised Independent Study. (1-4) **
Course may be repeated for credit. Enrollment is restricted; see the Introduction to Courses and Curricula section of this catalog. Independent study, weekly meeting with faculty.
Must be taken on a *passed/not passed* basis.
*Prerequisites: Restricted to freshmen and sophomores only. Consent of instructor.*
Supervised independent study by academically superior, lower division students. 3.3 GPA required and prior consent of instructor who is to supervise the study. A written proposal must be submitted to the department chair for pre-approval.
(F,SP)
*Staff*

*Upper Division Courses*

**C103. Introduction to Mathematical Economics. (4) **
Three hours of lecture per week.
Selected topics illustrating the application of mathematics to economic theory. This course is intended for upper-division students in Mathematics, Statistics, the Physical Sciences, and Engineering, and for economics majors with adequate mathematical preparation. No economic background is required. Also listed as Economics C103.
*Staff*

**104. Introduction to Analysis. (4) **
Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
*Prerequisites: 53 and 54.*
The real number system. Sequences, limits, and continuous functions in R and R. The concept of a metric space. Uniform convergence, interchange of limit operations. Infinite series. Mean value theorem and applications. The Riemann integral.
(F,SP)
*Staff*

**H104. Honors Introduction to Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Honors section corresponding to 104. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.

**105. Second Course in Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Differential calculus in Rn: the derivative as a linear map; the chain rule; inverse and implicit function theorems. Lebesgue integration on the line; comparison of Lebesgue and Riemann integrals. Convergence theorems. Fourier series, L2 theory. Fubini's theorem, change of variable.
(SP)

**110. Linear Algebra. (4) **
Three hours of lecture per week and an additional two hours of discussion at the discretion of the instructor. Six hours of lecture per week and an additional two hours of discussion at the discretion of the instructor.
*Prerequisites: 54 or a course with equivalent linear algebra content.*
Matrices, vector spaces, linear transformations, inner products, determinants. Eigenvectors. QR factorization. Quadratic forms and Rayleigh's principle. Jordan canonical form, applications. Linear functionals.
(F,SP)
*Staff*

**H110. Honors Linear Algebra. (4) **
Three hours of lecture per week.
*Prerequisites: 54 or a course with equivalent linear algebra content.*
Honors section corresponding to course 110 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
(SP)

**113. Introduction to Abstract Algebra. (4) **
Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
*Prerequisites: 54 or a course with equivalent linear algebra content.*
Sets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.
(F,SP)
*Staff*

**H113. Honors Introduction to Abstract Algebra. (4) **
Three hours of lecture per week.
*Prerequisites: 54 or a course with equivalent linear algebra content.*
Honors section corresponding to 113. Recommended for students who enjoy mathematics and are good at it. Greater emphasis on theory and challenging problems.
(F)

**114. Second Course in Abstract Algebra. (4) **
Three hours of lecture per week.
*Prerequisites: 110 and 113, or consent of instructor.*
Further topics on groups, rings, and fields not covered in Math 113. Possible topics include the Sylow Theorems and their applications to group theory; classical groups; abelian groups and modules over a principal ideal domain; algebraic field extensions; splitting fields and Galois theory; construction and classification of finite fields.
(SP)
*Staff*

**115. Introduction to Number Theory. (4) **
Three hours of lecture per week, and at the discretion of the instructor, an additional two hours of discussion per week.
*Prerequisites: 53 and 54.*
Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.
(F,SP)

**116. Cryptography. (4) **
Three hours of lecture per week, and at the discretion of the instructor, an additional two hours of discussion per week.
*Prerequisites: 55.*
Construction and analysis of simple cryptosystems, public key cryptography, RSA, signature schemes, key distribution, hash functions, elliptic curves, and applications.
(F,SP)

**118. Fourier Analysis, Wavelets, and Signal Processing. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Introduction to signal processing including Fourier analysis and wavelets. Theory, algorithms, and applications to one-dimensional signals and multidimensional images.
(F,SP)

**121A. Mathematical Tools for the Physical Sciences. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Rapid review of series and partial differentiation, complex variables and analytic functions, integral transforms, calculus of variations.

**121B. Mathematical Tools for the Physical Sciences. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Intended for students in the physical sciences who are not planning to take more advanced mathematics courses. Special functions, series solutions of ordinary differential equations, partial differential equations arising in mathematical physics, probability theory.

**123. Ordinary Differential Equations. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Existence and uniqueness of solutions, linear systems, regular singular points. Other topics selected from analytic systems, autonomous systems, Sturm-Liouville Theory.
(F)

**125A. Mathematical Logic. (4) **
Three hours of lecture per week.
*Prerequisites: Math 113 or consent of instructor.*
Sentential and quantificational logic. Formal grammar, semantical interpretation, formal deduction, and their interrelation. Applications to formalized mathematical theories. Selected topics from model theory or proof theory.
(F,SP)

**126. Introduction to Partial Differential Equations. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Waves and diffusion, initial value problems for hyperbolic and parabolic equations, boundary value problems for elliptic equations, Green's functions, maximum principles, a priori bounds, Fourier transform.
(SP)

**127. Mathematical and Computational Methods in Molecular Biology. (4) **
Three hours of lecture per week.
*Prerequisites: 53, 54, and 55; Statistics 20 recommended.*
Introduction to mathematical and computational problems arising in the context of molecular biology. Theory and applications of combinatorics, probability, statistics, geometry, and topology to problems ranging from sequence determination to structure analysis.
(F,SP)

**128A. Numerical Analysis. (4) **
Three hours of lecture and one hour of discussion per week. At the discretion of instructor, an additional hour of discussion/computer laboratory per week.
*Prerequisites: 53 and 54.*
Programming for numerical calculations, round-off error, approximation and interpolation, numerical quadrature, and solution of ordinary differential equations. Practice on the computer.
(F,SP)

**128B. Numerical Analysis. (4) **
Three hours of lecture and one hour of discussion per week. At the discretion of the instructor, an additional hour of discussion/computer laboratory per week.
*Prerequisites: 110 and 128A.*
Iterative solution of systems of nonlinear equations, evaluation of eigenvalues and eigenvectors of matrices, applications to simple partial differential equations. Practice on the computer.
(F,SP)

**130. The Classical Geometries. (4) **
Three hours of lecture per week.
*Prerequisites: 110 and 113.*
A critical examination of Euclid's Elements; ruler and compass constructions; connections with Galois theory; Hilbert's axioms for geometry, theory of areas, introduction of coordinates, non-Euclidean geometry, regular solids, projective geometry.
(F,SP)

**135. Introduction to the Theory of Sets. (4) **
Three hours of lecture per week.
*Prerequisites: 113 and 104.*
Set-theoretical paradoxes and means of avoiding them. Sets, relations, functions, order and well-order. Proof by transfinite induction and definitions by transfinite recursion. Cardinal and ordinal numbers and their arithmetic. Construction of the real numbers. Axiom of choice and its consequences.
(F,SP)

**136. Incompleteness and Undecidability. (4) **
Three hours of lecture per week.
*Prerequisites: 53, 54, and 55.*
Functions computable by algorithm, Turing machines, Church's thesis. Unsolvability of the halting problem, Rice's theorem. Recursively enumerable sets, creative sets, many-one reductions. Self-referential programs. Godel's incompleteness theorems, undecidability of validity, decidable and undecidable theories.
(F,SP)

**140. Metric Differential Geometry. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Frenet formulas, isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet-Von Dyck Theorem.
(F,SP)

**141. Elementary Differential Topology. (4) **
Three hours of lecture per week.
*Prerequisites: 104 or equivalent and linear algebra.*
Manifolds in n-dimensional Euclidean space and smooth maps, Sard's Theorem, classification of compact one-manifolds, transversality and intersection modulo 2.

**142. Elementary Algebraic Topology. (4) **
Three hours of lecture per week.
*Prerequisites: 104 and 113.*
The topology of one and two dimensional spaces: manifolds and triangulation, classification of surfaces, Euler characteristic, fundamental groups, plus further topics at the discretion of the instructor.
(F)

**143. Elementary Algebraic Geometry. (4) **
Three hours of lecture per week.
*Prerequisites: 113.*
Introduction to basic commutative algebra, algebraic geometry, and computational techniques. Main focus on curves, surfaces and Grassmannian varieties.
(F,SP)

**151. Mathematics of the Secondary School Curriculum I. (4) **
Three hours of lecture and zero to one hour of discussion per week.
*Prerequisites: 1A-1B, 53, or equivalent.*
Theory of rational numbers based on the number line, the Euclidean algorithm and fractions in lowest terms. The concepts of congruence and similarity, equation of a line, functions, and quadratic functions.
(F,SP)
*Staff*

**152. Mathematics of the Secondary School Curriculum II. (4) **
Three hours of lecture and zero to one hour of discussion per week.
*Prerequisites: 151; 54, 113, or equivalent.*
Complex numbers and Fundamental Theorem of Algebra, roots and factorizations of polynomials, Euclidean geometry and axiomatic systems, basic trigonometry.
(F,SP)
*Staff*

**153. Mathematics of the Secondary School Curriculum III. (4) **
Three hours of lecture and zero to one hour of discussion per week.
*Prerequisites: 151, 152.*
The real line and least upper bound, limit and decimal expansion of a number, differentiation and integration, Fundamental Theorem of Calculus, characterizations of sine, cosine, exp, and log.
(F,SP)
*Staff*

**160. History of Mathematics. (4) **
Three hours of lecture per week.
*Prerequisites: 53, 54, and 113.*
History of algebra, geometry, analytic geometry, and calculus from ancient times through the seventeenth century and selected topics from more recent mathematical history.
(SP)

**170. Mathematical Methods for Optimization. (4) **
Three hours of lecture per week.
*Prerequisites: 53 and 54.*
Linear programming and a selection of topics from among the following: matrix games, integer programming, semidefinite programming, nonlinear programming, convex analysis and geometry, polyhedral geometry, the calculus of variations, and control theory.
(F,SP)
*Staff*

**172. Combinatorics. (4) **
Three hours of lecture per week.
*Prerequisites: 55.*
Basic combinatorial principles, graphs, partially ordered sets, generating functions, asymptotic methods, combinatorics of permutations and partitions, designs and codes. Additional topics at the discretion of the instructor.
(F,SP)
*Staff*

**185. Introduction to Complex Analysis. (4) **
Three hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week. Six hours of lecture per week; at the discretion of the instructor, an additional two hours of discussion per week.
*Prerequisites: 104.*
Analytic functions of a complex variable. Cauchy's integral theorem, power series, Laurent series, singularities of analytic functions, the residue theorem with application to definite integrals. Some additional topics such as conformal mapping.
(F,SP)
*Staff*

**H185. Honors Introduction to Complex Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Honors section corresponding to Math 185 for exceptional students with strong mathematical inclination and motivation. Emphasis is on rigor, depth, and hard problems.
(SP)

**189. Mathematical Methods in Classical and Quantum Mechanics. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: 104, 110, 2 semesters lower division Physics.*
Topics in mechanics presented from a mathematical viewpoint: e.g., hamiltonian mechanics and symplectic geometry, differential equations for fluids, spectral theory in quantum mechanics, probability theory and statistical mechanics. See department bulletins for specific topics each semester course is offered.
(SP)

**191. Experimental Courses in Mathematics. (1-4) **
Course may be repeated for credit. Hours to be arranged.
*Prerequisites: Consent of instructor.*
The topics to be covered and the method of instruction to be used will be announced at the beginning of each semester that such courses are offered. See departmental bulletins.
(F,SP)

**195. Special Topics in Mathematics. (4) **
Course may be repeated for credit. Hours to be arranged.
*Prerequisites: Consent of instructor.*
Lectures on special topics, which will be announced at the beginning of each semester that the course is offered.

**196. Honors Thesis. (4) **
Course may be repeated for credit. Hours to be arranged.
*Prerequisites: Admission to the Honors Program; an overall GPA of 3.3 and a GPA of 3.5 in the major.*
Independent study of an advanced topic leading to an honors thesis.
(F,SP)

**197. Field Study. (1-4) **
Course may be repeated for credit. Three hours of work per week per unit.
Must be taken on a *passed/not passed* basis.
*Prerequisites: Upper division standing. Written proposal signed by faculty sponsor and approved by department chair.*
For Math/Applied math majors. Supervised experience relevant to specific aspects of their mathmatical emphasis of study in off-campus organizations. Regular individual meetings with faculty sponsor and written reports required. Units will be awarded on the basis of three hours/week/unit.
(F,SP)
*Staff*

**198. Directed Group Study. (1-4) **
Group study.
Must be taken on a *passed/not passed* basis.
*Prerequisites: Must have completed 60 units and be in good standing.*
Topics will vary with instructor.
(F,SP)
*Staff*

**198BC. Berkeley Connect. (1) **
Course may be repeated for credit. One hour of discussion per week.
Must be taken on a passed/not passed basis.
Berkeley Connect is a mentoring program, offered through various academic departments, that helps students build intellectual community. Over the course of a semester, enrolled students participate in regular small-group discussions facilitated by a graduate student mentor (following a faculty-directed curriculum), meet with their graduate student mentor for one-on-one academic advising, attend lectures and panel discussions featuring department faculty and alumni, and go on field trips to campus resources. Students are not required to be declared majors in order to participate.
(F,SP)

**199. Supervised Independent Study and Research. (1-4) **
Hours to be arranged.
Must be taken on a *passed/not passed* basis.
*Prerequisites: The standard college regulations for all 199 courses.*
(F,SP)

*Graduate Courses*

**202A. Introduction to Topology and Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Metric spaces and general topological spaces. Compactness and connectedness. Characterization of compact metric spaces. Theorems of Tychonoff, Urysohn, Tietze. Complete spaces and the Baire category theorem. Function spaces; Arzela-Ascoli and Stone-Weierstrass theorems. Partitions of unity. Locally compact spaces; one-point compactification. Introduction to measure and integration. Sigma algebras of sets. Measures and outer measures. Lebesgue measure on the line and Rn. Construction of the integral. Dominated convergence theorem.
(F,SP)

**202B. Introduction to Topology and Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 202A and 110.*
Measure and integration. Product measures and Fubini-type theorems. Signed measures; Hahn and Jordan decompositions. Radon-Nikodym theorem. Integration on the line and in Rn. Differentiation of the integral. Hausdorff measures. Fourier transform. Introduction to linear topological spaces, Banach spaces and Hilbert spaces. Banach-Steinhaus theorem; closed graph theorem. Hahn-Banach theorem. Duality; the dual of LP. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. Additional topics chosen may include compact operators, spectral theory of compact operators, and applications to integral equations.
(F,SP)

**203. Asymptotic Analysis in Applied Mathematics. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Asymptotic methods for differential equations, with emphasis upon many physical examples. Topics will include matched asymptotic expansions, Laplace's method, stationary phase, boundary layers, multiple scales, WKB approximations, asymptotic Lagrangians, bifurcation theory.
(F,SP)

**204. Ordinary Differential Equations. (4) **
Three hours of lecture per week.
*Prerequisites: 104.*
Rigorous theory of ordinary differential equations. Fundamental existence theorems for initial and boundary value problems, variational equilibria, periodic coefficients and Floquet Theory, Green's functions, eigenvalue problems, Sturm-Liouville theory, phase plane analysis, Poincare-Bendixon Theorem, bifurcation, chaos.
(F,SP)

**205. Theory of Functions of a Complex Variable. (4) **
Three hours of lecture per week.
*Prerequisites: 185.*
Normal families. Riemann Mapping Theorem. Picard's theorem and related theorems. Multiple-valued analytic functions and Riemann surfaces. Further topics selected by the instructor may include: harmonic functions, elliptic and algebraic functions, boundary behavior of analytic functions and HP spaces, the Riemann zeta functions, prime number theorem.

**206. Banach Algebras and Spectral Theory. (4) **
Three hours of lecture per week.
*Prerequisites: 202A-202B.*
Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. Analytic functional calculus. Hilbert space operators. C*-algebras of operators. Commutative C*-algebras. Spectral theorem for bounded self-adjoint and normal operators (both forms: the spectral integral and the "multiplication operator" formulation). Riesz theory of compact operators. Hilbert-Schmidt operators. Fredholm operators. The Fredholm index. Selected additional topics.
(F)

**207. Unbounded Operators. (4) **
Three hours of lecture per week.
*Prerequisites: 206.*
Unbounded self-adjoint operators. Stone's Theorem, Friedrichs extensions. Examples and applications, including differential operators. Perturbation theory. Further topics may include: unbounded operators in quantum mechanics, Stone-Von Neumann Theorem. Operator semigroups and evolution equations, some non-linear operators. Weyl theory of defect indices for ordinary differential operators.

**208. C*-algebras. (4) **
Three hours of lecture per week.
*Prerequisites: 206.*
Basic theory of C*-algebras. Positivity, spectrum, GNS construction. Group C*-algebras and connection with group representations. Additional topics, for example, C*-dynamical systems, K-theory.

**209. Von Neumann Algebras. (4) **
Three hours of lecture per week.
*Prerequisites: 206.*
Basic theory of von Neumann algebras. Density theorems, topologies and normal maps, traces, comparison of projections, type classification, examples of factors. Additional topics, for example, Tomita Takasaki theory, subfactors, group actions, and noncommutative probability.

**212. Several Complex Variables. (4) **
Three hours of lecture per week.
*Prerequisites: 185 and 202A-202B or their equivalents.*
Power series developments, domains of holomorphy, Hartogs' phenomenon, pseudo convexity and plurisubharmonicity. The remainder of the course may treat either sheaf cohomology and Stein manifolds, or the theory of analytic subvarieties and spaces.

**214. Differentiable Manifolds. (4) **
Three hours of lecture per week.
*Prerequisites: 202A.*
Smooth manifolds and maps, tangent and normal bundles. Sard's theorem and transversality, Whitney embedding theorem. Morse functions, differential forms, Stokes' theorem, Frobenius theorem. Basic degree theory. Flows, Lie derivative, Lie groups and algebras. Additional topics selected by instructor.
(F,SP)

**215A-215B. Algebraic Topology. (4;4) **
Three hours of lecture per week.
*Prerequisites: 113 and point-set topology (e.g. 202A).*
Fundamental group and covering spaces, simplicial and singular homology theory with applications, cohomology theory, duality theorem. Homotopy theory, fibrations, relations between homotopy and homology, obstruction theory, and topics from spectral sequences, cohomology operations, and characteristic classes. Sequence begins fall.

**C218A. Probability Theory. (4) **
Three hours of lecture per week.
The course is designed as a sequence with Statistics C205B/Mathematics C218B with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Also listed as Statistics C205A.
*Staff*

**C218B. Probability Theory. (4) **
Three hours of lecture per week.
The course is designed as a sequence with with Statistics C205A/Mathematics C218A with the following combined syllabus. Measure theory concepts needed for probability. Expection, distributions. Laws of large numbers and central limit theorems for independent random variables. Characteristic function methods. Conditional expectations, martingales and martingale convergence theorems. Markov chains. Stationary processes. Brownian motion. Also listed as Statistics C205B.
*Staff*

**219. Dynamical Systems. (4) **
Three hours of lecture per week.
*Prerequisites: 214.*
Diffeomorphisms and flows on manifolds. Ergodic theory. Stable manifolds, generic properties, structural stability. Additional topics selected by the instructor.
(F)

**220. Introduction to Probabilistic Methods in Mathematics and the Sciences. (4) **
Three hours of lecture per week.
*Prerequisites: Some familiarity with differential equations and their applications.*
Brownian motion, Langevin and Fokker-Planck equations, path integrals and Feynman diagrams, time series, an introduction to statistical mechanics, Monte Carlo methods, selected applications.
(F,SP)

**221. Advanced Matrix Computations. (4) **
Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Direct solution of linear systems, including large sparse systems: error bounds, iteration methods, least square approximation, eigenvalues and eigenvectors of matrices, nonlinear equations, and minimization of functions.
(F,SP)

**222A. Partial Differential Equations. (4) **
Three hours of lecture per week.
*Prerequisites: 105 or 202A.*
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Laplace's equation, heat equation, wave equation, nonlinear first-order equations, conservation laws, Hamilton-Jacobi equations, Fourier transform, Sobolev spaces.
(F)

**222B. Partial Differential Equations. (4) **
Three hours of lecture per week.
*Prerequisites: 105 or 202A.*
The theory of boundary value and initial value problems for partial differential equations, with emphasis on nonlinear equations. Second-order elliptic equations, parabolic and hyperbolic equations, calculus of variations methods, additional topics selected by instructor.
(F)

**C223A. Advanced Topics in Probability and Stochastic Process. (3) **
Course may be repeated for credit with a different instructor. Three hours of lecture per week.
*Prerequisites: Statistics C205A-C205B.*
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probabilty offered according to students demand and faculty availability. Also listed as Statistics C206A.
(F,SP)
*Staff*

**C223B. Advanced Topics in Probablity and Stochastic Processes. (3) **
Course may be repeated for credit with a different instructor. Three hours of lecture per week.
*Prerequisites: Statistics C205A-C205B.*
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probablity offered according to students demand and faculty availability. Also listed as Statistics C206B.
(F,SP)
*Staff*

**224A-224B. Mathematical Methods for the Physical Sciences. (4;4) **
Three hours of lecture per week.
*Prerequisites: Graduate status or consent of instructor.*
Introduction to the theory of distributions. Fourier and Laplace transforms. Partial differential equations. Green's function. Operator theory, with applications to eigenfunction expansions, perturbation theory and linear and non-linear waves. Sequence begins fall.
(F,SP)

**225A-225B. Metamathematics. (4;4) **
Three hours of lecture per week.
*Prerequisites: 125B and 135.*
Metamathematics of predicate logic. Completeness and compactness theorems. Interpolation theorem, definability, theory of models. Metamathematics of number theory, recursive functions, applications to truth and provability. Undecidable theories. Sequence begins fall.

**227A. Theory of Recursive Functions. (4) **
Three hours of lecture per week.
*Prerequisites: 225B.*
Recursive and recursively enumerable sets of natural numbers; characterizations, significance, and classification. Relativization, degrees of unsolvability. The recursion theorem. Constructive ordinals, the hyperarithmetical and analytical hierarchies. Recursive objects of higher type. Sequence begins fall.

**228A-228B. Numerical Solution of Differential Equations. (4;4) **
Three hours of lecture per week.
*Prerequisites: 128A.*
Ordinary differential equations: Runge-Kutta and predictor-corrector methods; stability theory, Richardson extrapolation, stiff equations, boundary value problems. Partial differential equations: stability, accuracy and convergence, Von Neumann and CFL conditions, finite difference solutions of hyperbolic and parabolic equations. Finite differences and finite element solution of elliptic equations.

**229. Theory of Models. (4) **
Three hours of lecture per week.
*Prerequisites: 225B.*
Syntactical characterization of classes closed under algebraic operations. Ultraproducts and ultralimits, saturated models. Methods for establishing decidability and completeness. Model theory of various languages richer than first-order.

**235A. Theory of Sets. (4) **
Three hours of lecture per week.
*Prerequisites: 125A and 135.*
Axiomatic foundations. Operations on sets and relations. Images and set functions. Ordering, well-ordering, and well-founded relations; general principles of induction and recursion. Ranks of sets, ordinals and their arithmetic. Set-theoretical equivalence, similarity of relations; definitions by abstraction. Arithmetic of cardinals. Axiom of choice, equivalent forms, and consequences. Sequence begins fall.
(SP)
*125A and 135.*

**236. Metamathematics of Set Theory. (4) **
Three hours of lecture per week.
*Prerequisites: 225B and 235A.*
Various set theories: comparison of strength, transitive, and natural models, finite axiomatizability. Independence and consistency of axiom of choice, continuum hypothesis, etc. The measure problem and axioms of strong infinity.

**239. Discrete Mathematics for the Life Sciences. (4) **
Three hours of lecture per week.
*Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor.*
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry.
(F,SP)
*Staff*

**C239. Discrete Mathematics for the Life Sciences. (4) **
Three hours of lecture per week.
*Prerequisites: Statistics 134 or equivalent introductory probability theory course, or consent of instructor.*
Introduction to algebraic statistics and probability, optimization, phylogenetic combinatorics, graphs and networks, polyhedral and metric geometry. Also listed as Molecular and Cell Biology C244.
(F,SP)
*Staff*

**240. Riemannian Geometry. (4) **
Three hours of lecture per week.
*Prerequisites: 214.*
Riemannian metric and Levi-Civita connection, geodesics and completeness, curvature, first and second variations of arc length. Additional topics such as the theorems of Myers, Synge, and Cartan-Hadamard, the second fundamental form, convexity and rigidity of hypersurfaces in Euclidean space, homogeneous manifolds, the Gauss-Bonnet theorem, and characteristic classes.
(SP)

**241. Complex Manifolds. (4) **
Three hours of lecture per week.
*Prerequisites: 214 and 215A.*
Riemann surfaces, divisors and line bundles on Riemann surfaces, sheaves and the Dolbeault theorem on Riemann surfaces, the classical Riemann-Roch theorem, theorem of Abel-Jacobi. Complex manifolds, Kahler metrics. Summary of Hodge theory, groups of line bundles, additional topics such as Kodaira's vanishing theorem, Lefschetz hyperplane theorem.
(SP)

**242. Symplectic Geometry. (4) **
Three hours of lecture per week.
*Prerequisites: 214.*
Basic topics: symplectic linear algebra, symplectic manifolds, Darboux theorem, cotangent bundles, variational problems and Legendre transform, hamiltonian systems, Lagrangian submanifolds, Poisson brackets, symmetry groups and momentum mappings, coadjoint orbits, Kahler manifolds.
(F,SP)

**C243. Seq: Methods and Applications. (3) **
Three hours of lecture per week.
*Prerequisites: Graduate standing in Math, MCB, and Computational Biology; or consent of the instructor.*
A graduate seminar class in which a group of students will closely examine recent computational methods in high-throughput sequencing followed by directly examining interesting biological applications thereof. Also listed as Molecular and Cell Biology C243.
(SP)
*Pachter*

**245A. General Theory of Algebraic Structures. (4) **
Three hours of lecture per week.
*Prerequisites: Math 113.*
Structures defined by operations and/or relations, and their homomorphisms. Classes of structures determined by identities. Constructions such as free objects, objects presented by generators and relations, ultraproducts, direct limits. Applications of general results to groups, rings, lattices, etc. Course may emphasize study of congruence- and subalgebra-lattices, or category-theory and adjoint functors, or other aspects.
(F,SP)

**249. Algebraic Combinatorics. (4) **
Three hours of lecture per week.
*Prerequisites: 250A or consent of instructor.*
(I) Enumeration, generating functions and exponential structures, (II) Posets and lattices, (III) Geometric combinatorics, (IV) Symmetric functions, Young tableaux, and connections with representation theory. Further study of applications of the core material and/or additional topics, chosen by instructor.
(F,SP)
*Staff*

**250A. Groups, Rings, and Fields. (4) **
Three hours of lecture per week.
*Prerequisites: 114 or consent of instructor.*
Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.
(F)

**250B. Multilinear Algebra and Further Topics. (4) **
Three hours of lecture per week.
*Prerequisites: 250A.*
Tensor algebras and exterior algebras, with application to linear transformations. Commutative ideal theory, localization. Elementary specialization and valuation theory. Related topics in algebra.
(SP)

**251. Ring Theory. (4) **
Three hours of lecture per week.
*Prerequisites: 250A.*
Topics such as: Noetherian rings, rings with descending chain condition, theory of the radical, homological methods.

**252. Representation Theory. (4) **
Three hours of lecture per week.
*Prerequisites: 250A.*
Structure of finite dimensional algebras, applications to representations of finite groups, the classical linear groups.
(F)

**253. Homological Algebra. (4) **
Three hours of lecture per week.
*Prerequisites: 250A.*
Modules over a ring, homomorphisms and tensor products of modules, functors and derived functors, homological dimension of rings and modules.

**254A-254B. Number Theory. (4;4) **
254B may be repeated with consent of instructor. Three hours of lecture per week.
*Prerequisites: 250A for 254A; 254A for 254B.*
Valuations, units, and ideals in number fields, ramification theory, quadratic and cyclotomic fields, topics from class field theory, zeta-functions and L-series, distribution of primes, modular forms, quadratic forms, diophantine equations, P-adic analysis, and transcendental numbers. Sequence begins fall.

**255. Algebraic Curves. (4) **
Three hours of lecture per week.
*Prerequisites: 250A-250B or consent of instructor.*
Elliptic curves. Algebraic curves, Riemann surfaces, and function fields. Singularities. Riemann-Roch theorem, Hurwitz's theorem, projective embeddings and the canonical curve. Zeta functions of curves over finite fields. Additional topics such as Jacobians or the Riemann hypothesis.
(F,SP)

**256A-256B. Algebraic Geometry. (4;4) **
Three hours of lecture per week.
*Prerequisites: 250A-250B for 256A; 256A for 256B.*
Affine and projective algebraic varieties. Theory of schemes and morphisms of schemes. Smoothness and differentials in algebraic geometry. Coherent sheaves and their cohomology. Riemann-Roch theorem and selected applications. Sequence begins fall.

**257. Group Theory. (4) **
Three hours of lecture per week.
*Prerequisites: 250A.*
Topics such as: generators and relations, infinite discrete groups, groups of Lie type, permutation groups, character theory, solvable groups, simple groups, transfer and cohomological methods.

**258. Classical Harmonic Analysis. (4) **
Three hours of lecture per week.
*Prerequisites: 206 or a basic knowledge of real, complex, and linear analysis.*
Basic properties of Fourier series, convergence and summability, conjugate functions, Hardy spaces, boundary behavior of analytic and harmonic functions. Additional topics at the discretion of the instructor.

**261A-261B. Lie Groups. (4;4) **
Three hours of lecture per week.
*Prerequisites: 214.*
Lie groups and Lie algebras, fundamental theorems of Lie, general structure theory; compact, nilpotent, solvable, semi-simple Lie groups; classification theory and representation theory of semi-simple Lie algebras and Lie groups, further topics such as symmetric spaces, Lie transformation groups, etc., if time permits. In view of its simplicity and its wide range of applications, it is preferable to cover compact Lie groups and their representations in 261A. Sequence begins Fall.

**265. Differential Topology. (4) **
Three hours of lecture per week.
*Prerequisites: 214 plus 215A or some familiarity with algebraic topology.*
Approximations, degrees of maps, vector bundles, tubular neighborhoods. Introduction to Morse theory, handlebodies, cobordism, surgery. Additional topics selected by instructor from: characteristic classes, classification of manifolds, immersions, embeddings, singularities of maps.

**270. Hot Topics Course in Mathematics. (2) **
Course may be repeated for credit as topic varies. One and one-half hours of lecture per week.
Must be taken on a *satisfactory/unsatisfactory* basis.
This course will give introductions to current research developments. Every semester we will pick a different topic and go through the relevant literature. Each student will be expected to give one presentation.
(F,SP)
*Staff*

**273. Topics in Numerical Analysis. **
Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.
(F,SP)

273A. Ordinary Differential Equations. (4)

273F. Topics in Computational Physics. (4)

273I. Approximation Theory. (4)

**274. Topics in Algebra. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**275. Topics in Applied Mathematics. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**276. Topics in Topology. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**277. Topics in Differential Geometry. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**278. Topics in Analysis. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**279. Topics in Partial Differential Equations. (4) **
Course may be repeated for credit. Three hours of lecture per week.
*Prerequisites: Consent of instructor.*
Advanced topics chosen by the instructor. The content of this course changes, as in the case of seminars.

**290. Seminars. (1-6) **
Course may be repeated for credit. Hours to be arranged.
Topics in foundations of mathematics, theory of numbers, numerical calculations, analysis, geometry, topology, algebra, and their applications, by means of lectures and informal conferences; work based largely on original memoirs.
(F,SP)

**295. Individual Research. (1-12) **
Course may be repeated for credit. Hours to be arranged.
Sections 1-30 to be graded on a letter-grade basis. Sections 31-60 to be graded on a *satisfactory/unsatisfactory* basis.
Intended for candidates for the Ph.D. degree.
(F,SP)

**299. Reading Course for Graduate Students. (1-6) **
Course may be repeated for credit. Hours to be arranged.
Sections 1-30 to be graded on a letter-grade basis. Sections 31-60 to be graded on a *satisfactory/unsatisfactory* basis.
Investigation of special problems under the direction of members of the department.
(F,SP)

**600. Individual Study for Master's Students. (1-6) **
Course may be repeated for credit. Course does not satisfy unit or residence requirements for master's degree. Hours to be arranged.
Must be taken on a *satisfactory/unsatisfactory* basis.
*Prerequisites: For candidates for master's degree.*
Individual study for the comprehensive or language requirements in consultation with the field adviser.
(F,SP)

**602. Individual Study for Doctoral Students. (1-8) **
Course may be repeated for credit.
Must be taken on a *satisfactory/unsatisfactory* basis.
*Prerequisites: For qualified graduate students.*
Individual study in consultation with the major field adviser intended to provide an opportunity for qualified students to prepare themselves for the various examinations required for candidates for the Ph.D. Course does not satisfy unit or residence requirements for doctoral degree.
(F,SP)
*Staff*

*Professional Courses*

**301. Undergraduate Mathematics Instruction. (1-2) **
Course may be repeated once for credit. Three hours of seminar and four hours of tutorial per week.
Must be taken on a *passed/not passed* basis.
*Prerequisites: Permission of SLC instructor, as well as sophomore standing and at least a B average in two semesters of calculus. Apply at Student Learning Center.*
May be taken for one unit by special permission of instructor. Tutoring at the Student Learning Center or for the Professional Development Program.
(F,SP)

**303. Professional Preparation: Supervised Teaching of Mathematics. (2-4) **
Course may be repeated for credit. Two to four hours of independent study per week.
Must be taken on a satisfactory/unsatisfactory basis.
*Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor.*
Meeting with supervising faculty and with discussion sections. Experience in teaching under the supervision of Mathematics faculty.
(F,SP)
*Staff*

**375. Teaching Workshop. (4) **
Two hours of lecture per week, plus class visits.
Must be taken on a *satisfactory/unsatisfactory* basis.
*Prerequisites: 300, graduate standing and appointment as a Graduate Student Instructor.*
*Formerly Mathematics 300.*
Mandatory for all graduate student instructors teaching for the first time in the Mathematics Department. The course consists of practice teaching, alternatives to standard classroom methods, guided group and self-analysis of videotapes, reciprocal classroom visitations, and an individual project.
(F,SP)
*Staff*